Search results for "Combinatorics"
showing 10 items of 1770 documents
The Zero-Check for Eliminating Non-Significant Elements
1970
During continued matrix operations like the simplex method a lot of small non-significant elements, the actual value of which is zero,usually augment the working coefficient matrix. These elements are caused by round-off errors. They arise in the following manner in a computation of the type: $${\rm{d}}\, = \,{\rm{a}}\,{\rm{ - }}\,{\rm{b}}{\rm{.c}}$$ with e.g. the data (in FORTRAN notation) $${\rm{a}}\, = \,{\rm{2}}\,{\rm{ = }}\,{\rm{.20000000}}\,{\rm{E}}\,{\rm{01}}$$ $${\rm{b}}\, = \,{\rm{6}}\,{\rm{ = }}\,{\rm{.60000000}}\,{\rm{E}}\,{\rm{01}}$$ $${\rm{c}}\, = \,{\rm{1/3}}\,{\rm{ = }}\,{\rm{.33333333}}\,{\rm{E}}\,{\rm{00}}$$
Commutator Equations and the Equationally-Defined Commutator
2015
If \(\underline{s} = s_{1},\ldots,s_{m}\), and \(\underline{t} = t_{1},\ldots,t_{m}\) are sequences of terms (both sequences of the same length m) and X is a set of equations then $$\displaystyle{\underline{s} \approx \underline{ t} \in X}$$ abbreviates the fact that s i ≈ t i ∈ X for i = 1, …, m.)
Canonical Adiabatic Theory
2001
In the present chapter we are concerned with systems, the change of which—with the exception of a single degree of freedom—should proceed slowly. (Compare the pertinent remarks about \(\varepsilon\) as slow parameter in Chap. 7) Accordingly, the Hamiltonian reads: $$\displaystyle{ H = H_{0}{\bigl (J,\varepsilon p_{i},\varepsilon q_{i};\varepsilon t\bigr )} +\varepsilon H_{1}{\bigl (J,\theta,\varepsilon p_{i},\varepsilon q_{i};\varepsilon t\bigr )}\;. }$$ (12.1) Here, \((J,\theta )\) designates the “fast” action-angle variables for the unperturbed, solved problem \(H_{0}(\varepsilon = 0),\) and the (p i , q i ) represent the remaining “slow” canonical variables, which do not necessarily have…
Particle in Harmonic E-Field E(t) = Esinω 0 t; Schwinger–Fock Proper-Time Method
2017
Since the Green’s function of a Dirac particle in an external field, which is described by a potential A μ (x), is given by $$\displaystyle{ \left [\gamma \cdot \left (\frac{1} {i} \partial - eA\right ) + m\right ]G(x,x^{{\prime}}\vert A) =\delta (x - x^{{\prime}}) }$$ (37.1) the Green operator G+[A] is defined by $$\displaystyle{ \left (\gamma \Pi + m\right )G_{+} = 1\,,\quad \Pi _{\mu } = p_{\mu } - eA_{\mu } }$$ or $$\displaystyle\begin{array}{rcl} G_{+}& =& \frac{1} {\gamma \Pi + m - i\epsilon }\,,\quad \epsilon > 0 {}\\ & =& \frac{\gamma \Pi - m} {\left (\gamma \Pi \right )^{2} - m^{2} + i\epsilon } = \frac{-\gamma \Pi + m} {m^{2} -\left (\gamma \Pi \right )^{2} - i\epsilon } {}\\ & =&…
Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators
2017
Let \begin{document}$A∈{\rm{Sym}}(n× n)$\end{document} be an elliptic 2-tensor. Consider the anisotropic fractional Schrodinger operator \begin{document}$\mathscr{L}_A^s+q$\end{document} , where \begin{document}$\mathscr{L}_A^s: = (-\nabla·(A(x)\nabla))^s$\end{document} , \begin{document}$s∈ (0, 1)$\end{document} and \begin{document}$q∈ L^∞$\end{document} . We are concerned with the simultaneous recovery of \begin{document}$q$\end{document} and possibly embedded soft or hard obstacles inside \begin{document}$q$\end{document} by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain \begin{document}$Ω$\end{document} associated with \begin{document}$\mathscr{L}_A^s+q$\end{docume…
Glassy dynamics in confinement: planar and bulk limits of the mode-coupling theory.
2014
We demonstrate how the matrix-valued mode-coupling theory of the glass transition and glassy dynamics in planar confinement converges to the corresponding theory for two-dimensional (2D) planar and the three-dimensional bulk liquid, provided the wall potential satisfies certain conditions. Since the mode-coupling theory relies on the static properties as input, the emergence of a homogeneous limit for the matrix-valued intermediate scattering functions is directly connected to the convergence of the corresponding static quantities to their conventional counterparts. We show that the 2D limit is more subtle than the bulk limit, in particular, the in-planar dynamics decouples from the motion …
Diverging exchange force and form of the exact density matrix functional
2019
For translationally invariant one-band lattice models, we exploit the ab initio knowledge of the natural orbitals to simplify reduced density matrix functional theory (RDMFT). Striking underlying features are discovered: First, within each symmetry sector, the interaction functional $\mathcal{F}$ depends only on the natural occupation numbers $\bf{n}$. The respective sets $\mathcal{P}^1_N$ and $\mathcal{E}^1_N$ of pure and ensemble $N$-representable one-matrices coincide. Second, and most importantly, the exact functional is strongly shaped by the geometry of the polytope $\mathcal{E}^1_N \equiv \mathcal{P}^1_N $, described by linear constraints $D^{(j)}(\bf{n})\geq 0$. For smaller systems,…
Efficiencies of logical Bell measurements on Calderbank-Shor-Steane codes with static linear optics
2019
We show how the efficiency of a logical Bell measurement (BM) can be calculated for arbitrary Calderbank-Shor-Steane (CSS) codes with the experimentally important constraint of using only transversal static linear-optical BMs on the physical single-photon qubit level. For this purpose, we utilize the codes' description in terms of stabilizers in order to calculate general efficiencies for the loss-free case, but also for specific cases including photon loss. These efficiencies can be, for instance, used for obtaining transmission rates of all-optical quantum repeaters. In the loss-free case, we demonstrate that the important class of CSS codes with identical physical-qubit support for the t…
Feynman graph polynomials
2010
The integrand of any multi-loop integral is characterised after Feynman parametrisation by two polynomials. In this review we summarise the properties of these polynomials. Topics covered in this article include among others: Spanning trees and spanning forests, the all-minors matrix-tree theorem, recursion relations due to contraction and deletion of edges, Dodgson's identity and matroids.
Experiments on the Parallel Hall Effect in Three-Dimensional Metamaterials
2017
The usual Hall effect in a semiconductor leads to a voltage perpendicular to an applied static magnetic field. The authors significantly extend their recent work and demonstrate $e\phantom{\rule{0}{0ex}}x\phantom{\rule{0}{0ex}}p\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}m\phantom{\rule{0}{0ex}}e\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}l\phantom{\rule{0}{0ex}}y$ that not only the sign but also the direction of the Hall field can be tailored by a metamaterial's microstructure. They show that, with judicious engineering, the Hall voltage can be $p\phantom{\rule{0}{0…